Separation for the stationary Prandtl equation

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1 Separation for the stationary Prandtl equation Anne-Laure Dalibard (UPMC) with Nader Masmoudi (Courant Institute, NYU) February 13th-17th, 217 Dynamics of Small Scales in Fluids ICERM, Brown University

2 Outline Introduction Behaviour near separation: heuristics and formal results Main result and ideas Conclusion and perspectives

3 Introduction Plan Introduction Behaviour near separation: heuristics and formal results Main result and ideas Conclusion and perspectives

4 Introduction What is boundary layer separation? Flow with low viscosity around an obstacle, in the presence of an adverse pressure gradient (=the outer flow is decreasing along the obstacle): Figure: Cross-section of a flow past a cylinder (source: ONERA, France) After separation: widely open problem (turbulence?) Mathematical interest: understand the vanishing viscosity limit in the Navier-Stokes equation. Industrial interest: reduce drag.

5 Introduction Derivation of the (stationary) Prandtl system Starting point: stationary 2d Navier-Stokes with small viscosity ν 1 in a half-plane. Velocity field u ν : R 2 + R 2 with Ansatz: u ν (x, y) (u ν )u ν + p ν ν u ν = f in R 2 +, divu ν = in R 2 +, u ν y= =. (1) { u E (x, y) for y ( ( ) ν (sol. of 2d Euler), y u x, ν, ( )) y νv x, ν for y ν. Stationary Prandtl system: with Y = y/ ν, u x u + v Y u YY u = f 1 (x, ) pe (x, ) x x u + Y v =, u Y = =, v Y = =, lim Y u = ue (x, ).

6 Introduction Derivation of the (stationary) Prandtl system Starting point: stationary 2d Navier-Stokes with small viscosity ν 1 in a half-plane. Velocity field u ν : R 2 + R 2 with Ansatz: u ν (x, y) (u ν )u ν + p ν ν u ν = f in R 2 +, divu ν = in R 2 +, u ν y= =. (1) { u E (x, y) for y ( ( ) ν (sol. of 2d Euler), y u x, ν, ( )) y νv x, ν for y ν. Stationary Prandtl system: with Y = y/ ν, u x u + v Y u YY u = f 1 (x, ) pe (x, ) x x u + Y v =, u Y = =, v Y = =, lim Y u = ue (x, ).

7 Introduction Derivation of the (stationary) Prandtl system Starting point: stationary 2d Navier-Stokes with small viscosity ν 1 in a half-plane. Velocity field u ν : R 2 + R 2 with Ansatz: u ν (x, y) (u ν )u ν + p ν ν u ν = f in R 2 +, divu ν = in R 2 +, u ν y= =. (1) { u E (x, y) for y ( ( ) ν (sol. of 2d Euler), y u x, ν, ( )) y νv x, ν for y ν. Stationary Prandtl system: with Y = y/ ν, u x u + v Y u YY u = f 1 (x, ) pe (x, ) x x u + Y v =, u Y = =, v Y = =, lim Y u = ue (x, ).

8 Introduction The stationary Prandtl equation: general existence result Stationary Prandtl system: u x u + v Y u YY u = g(x) x u + Y v =, u x= = u u Y = =, v Y = =, lim Y u = ue (x, ). (P) Nonlocal, scalar evolution eq. in x. Locally well-posed as long as u > : Theorem [Oleinik, 1962]: Let u C 2,α b (R + ), α >. Assume that u (Y ) > for Y >, u () >, and that YY u + g() = O(Y 2 ) for < Y 1. Then there exists x > such that (P) has a unique strong C 2 solution in {(x, Y ) R 2, x < x, Y }. If g(x), then x = +.

9 Introduction The stationary Prandtl equation: general existence result Stationary Prandtl system: u x u + v Y u YY u = g(x) x u + Y v =, u x= = u u Y = =, v Y = =, lim Y u = ue (x, ). (P) Nonlocal, scalar evolution eq. in x. Locally well-posed as long as u > : Theorem [Oleinik, 1962]: Let u C 2,α b (R + ), α >. Assume that u (Y ) > for Y >, u () >, and that YY u + g() = O(Y 2 ) for < Y 1. Then there exists x > such that (P) has a unique strong C 2 solution in {(x, Y ) R 2, x < x, Y }. If g(x), then x = +.

10 Introduction The stationary Prandtl equation: monotonicity, comparison principle Nonlinear change of variables [von Mises]: transforms (P) into a local diffusion equation (porous medium type). Maximum principle holds for the new eq. by standard tools and arguments. Monotonicity is preserved by (P). Comparison principle for the Prandtl equation: Consider a super-solution ū for Prandtl; Von Mises sub/super solution w for the eq. in new variables; Maximum principle for the new eq.: w w. ODE arguments: u ū. Remark: we will consider strictly increasing solutions only: ensures that separation happens at the boundary.

11 Introduction General mechanism behind separation Setting: monotone solutions with adverse pressure gradient (g ). In general there exists x such that u =. Y x=x,y = For x > x, Y. 1, u takes negative values: reversed flow near the boundary. There exists a curve Y = F (x) such that u(x, F (x)) = : separation of the boundary layer. Definition: x is called the separation point.

12 Introduction Questions 1. Does separation really happen? Can you cook-up solutions of (P) such that Y u Y = (x) as x x for some finite x? 2. If you can, what is the rate at which Y u Y = (x) vanishes?

13 Introduction Related results Instability results (time dependent version): Local well-posedness in high regularity spaces (analytic, Gevrey) [Sammartino& Caflisch; Gérard-Varet& Masmoudi...] or for monotonic data [Oleinik; Masmoudi&Wong; Alexandre, Wang, Xu& Yang...] BUT instabilities develop in short time in Sobolev spaces [Grenier; Gérard-Varet&Dormy...] Formation of singularities (time dependent version): [Kukavica, Vicol, Wang](van Dommelen-Shen singularity) Starting from real analytic initial data, for specific outer Euler flow, some solutions display singularities in finite time. Justification of the Prandtl Ansatz when ν 1: [Guo& Nguyen, Iyer] Starting from stationary Navier-Stokes above a moving plate (non-zero boundary condition on the wall), local convergence/global convergence for small data ( no singularity).

14 Behaviour near separation: heuristics and formal results Plan Introduction Behaviour near separation: heuristics and formal results Main result and ideas Conclusion and perspectives

15 Behaviour near separation: heuristics and formal results Formal derivations of the self similar rate Formal computations of an exact solution to (P) by [Goldstein 48, Stewartson 58] thanks to Taylor expansions in self-similar variables. Self similar change of variables: rely on the observation that (P) is invariant by the scaling u(x, Y ) 1 µ u(µx, µ 1/4 Y ), v(x, y) µ 1/4 v(µx, µ 1/4 Y ), with µ >. Remark: the coefficients of the asymptotic expansion are never entirely determined (dependence on initial data?) Heuristic argument by Landau: Y u Y = (x) x x. (same as Goldstein& Stewartson.)

16 Behaviour near separation: heuristics and formal results Statement by Luis Caffarelli and Weinan E In a paper published in 2, Weinan E announces a joint result with Luis Caffarelli, stating: Theorem [Caffarelli, E, 1995]: Assume that g(x) = 1, and that u satisfies u 2 3 Y 2 Y u u. Then: There exists x > such that the solution cannot be extended beyond x ; The family u µ := 1 µ u(µ(x x), µ 1/4 Y ) is compact in C(R 2 +). The author also states two (non-trivial...) Lemmas playing a key role in the proof, relying on the maximum principle. Unfortunately the complete proof was never published... Goal of the present talk: propose an alternate proof, relying on different techniques, and giving a more quantitative result.

17 Behaviour near separation: heuristics and formal results Statement by Luis Caffarelli and Weinan E In a paper published in 2, Weinan E announces a joint result with Luis Caffarelli, stating: Theorem [Caffarelli, E, 1995]: Assume that g(x) = 1, and that u satisfies u 2 3 Y 2 Y u u. Then: There exists x > such that the solution cannot be extended beyond x ; The family u µ := 1 µ u(µ(x x), µ 1/4 Y ) is compact in C(R 2 +). The author also states two (non-trivial...) Lemmas playing a key role in the proof, relying on the maximum principle. Unfortunately the complete proof was never published... Goal of the present talk: propose an alternate proof, relying on different techniques, and giving a more quantitative result.

18 Main result and ideas Plan Introduction Behaviour near separation: heuristics and formal results Main result and ideas Conclusion and perspectives

19 Main result and ideas Main result Theorem [D., Masmoudi, 16]: Consider the equation (P) with g(x) = 1. Then for a class of initial data u = u x= satisfying u is strictly increasing with respect to Y ; u (Y ) λ Y + Y 2 2 for Y 1 and for some λ 1; separation occurs at a finite distance x = O(λ 2 ). Moreover for all x (, x ), λ(x) := Y u Y = (x) C x x and for some weight w = w(x, Y ), u u app L 2 (w) = o( u app L 2 (w)) as x x, where for Y (x x) 1/4 u app (x, Y ) = λ(x)y + Y 2 2 αy 4 βλ(x) 1 Y 7.

20 Main result and ideas Main result Theorem [D., Masmoudi, 16]: Consider the equation (P) with g(x) = 1. Then for a class of initial data u = u x= satisfying u is strictly increasing with respect to Y ; u (Y ) λ Y + Y 2 2 for Y 1 and for some λ 1; separation occurs at a finite distance x = O(λ 2 ). Moreover for all x (, x ), λ(x) := Y u Y = (x) C x x and for some weight w = w(x, Y ), u u app L 2 (w) = o( u app L 2 (w)) as x x, where for Y (x x) 1/4 u app (x, Y ) = λ(x)y + Y 2 2 αy 4 βλ(x) 1 Y 7.

21 Main result and ideas Remarks Two results: asymptotic behavior of Y u Y = and error estimate between u and u app. The self-similar rate is the one predicted by Landau, Goldstein and Stewartson. Existence of other (unstable) rates? Comparison with result by Caffarelli and E: Encompasses their result; More stringent assumptions; Quantitative result; the limit is identified. Tools and scheme of proof: Inspired by study of blow-up rates for NLS [Zakharov, Sulem& Sulem; Merle& Raphaël]; successfully applied to wave and Schrödinger maps, Keller-Segel system, harmonic heat flow [Merle, Raphaël, Rodnianski, Schweyer...] Perform a self-similar change of variables; approximate solution, energy estimates in rescaled variables; Use techniques based on modulation of variables to find the self-similar rate λ(x) = Y u Y =.

22 Main result and ideas Remarks Two results: asymptotic behavior of Y u Y = and error estimate between u and u app. The self-similar rate is the one predicted by Landau, Goldstein and Stewartson. Existence of other (unstable) rates? Comparison with result by Caffarelli and E: Encompasses their result; More stringent assumptions; Quantitative result; the limit is identified. Tools and scheme of proof: Inspired by study of blow-up rates for NLS [Zakharov, Sulem& Sulem; Merle& Raphaël]; successfully applied to wave and Schrödinger maps, Keller-Segel system, harmonic heat flow [Merle, Raphaël, Rodnianski, Schweyer...] Perform a self-similar change of variables; approximate solution, energy estimates in rescaled variables; Use techniques based on modulation of variables to find the self-similar rate λ(x) = Y u Y =.

23 Main result and ideas Remarks Two results: asymptotic behavior of Y u Y = and error estimate between u and u app. The self-similar rate is the one predicted by Landau, Goldstein and Stewartson. Existence of other (unstable) rates? Comparison with result by Caffarelli and E: Encompasses their result; More stringent assumptions; Quantitative result; the limit is identified. Tools and scheme of proof: Inspired by study of blow-up rates for NLS [Zakharov, Sulem& Sulem; Merle& Raphaël]; successfully applied to wave and Schrödinger maps, Keller-Segel system, harmonic heat flow [Merle, Raphaël, Rodnianski, Schweyer...] Perform a self-similar change of variables; approximate solution, energy estimates in rescaled variables; Use techniques based on modulation of variables to find the self-similar rate λ(x) = Y u Y =.

24 Main result and ideas The self-similar change of variables Let λ(x) := Y u Y = (x). Define ũ = ũ(x, ξ) by Then λ 4 ( ũũ x ũ ξ ξ Define s, b, U such that b = 2λ x λ 3, Then U satisfies ũ(x, ξ) = λ 2 (x)u(x, λ(x)ξ). ũ x ) + λ x λ 3 ( 2ũ 2 3ũ ξ ξ dx ds = λ4, ξ UU s U ξ U s bu 2 + 3b 2 U ξ ) ũ ũ ξξ = 1. U(s, ξ) = ũ(x(s), ξ). ξ U U ξξ = 1. Remark: x x corresponds to s provided x λ 4 =. (R)

25 Main result and ideas The self-similar change of variables Let λ(x) := Y u Y = (x). Define ũ = ũ(x, ξ) by Then λ 4 ( ũũ x ũ ξ ξ Define s, b, U such that b = 2λ x λ 3, Then U satisfies ũ(x, ξ) = λ 2 (x)u(x, λ(x)ξ). ũ x ) + λ x λ 3 ( 2ũ 2 3ũ ξ ξ dx ds = λ4, ξ UU s U ξ U s bu 2 + 3b 2 U ξ ) ũ ũ ξξ = 1. U(s, ξ) = ũ(x(s), ξ). ξ U U ξξ = 1. Remark: x x corresponds to s provided x λ 4 =. (R)

26 Main result and ideas The self-similar change of variables Let λ(x) := Y u Y = (x). Define ũ = ũ(x, ξ) by Then λ 4 ( ũũ x ũ ξ ξ Define s, b, U such that b = 2λ x λ 3, Then U satisfies ũ(x, ξ) = λ 2 (x)u(x, λ(x)ξ). ũ x ) + λ x λ 3 ( 2ũ 2 3ũ ξ ξ dx ds = λ4, ξ UU s U ξ U s bu 2 + 3b 2 U ξ ) ũ ũ ξξ = 1. U(s, ξ) = ũ(x(s), ξ). ξ U U ξξ = 1. Remark: x x corresponds to s provided x λ 4 =. (R)

27 Main result and ideas Strategy From now on, work only on equation on U: ξ UU s U ξ U s bu 2 + 3b 2 U ξ ξ U U ξξ = 1. At this stage, b is an unknown. The asymptotic behavior of b dictates the self-similar rate λ(x). Scheme of proof: 1. Construct an approximate solution; 2. Choose the approximate solution with the least possible growth at infinity: heuristics for the modulation rate b; 3. Energy estimate on the remainder of the solution. Rule of thumb: the expected rate λ(x) = C x x corresponds to b(s) = 1 s b s + b 2 =.

28 Plan Introduction Behaviour near separation: heuristics and formal results Main result and ideas Conclusion and perspectives

29 Plan Introduction Behaviour near separation: heuristics and formal results Main result and ideas The approximate solution Energy estimates Conclusion and perspectives

30 Construction of an approximate solution ξ ξ UU s U ξ U s bu 2 + 3b 2 U ξ U U ξξ = 1. }{{} =:A(U) Boundary conditions at ξ = : by definition of λ, U ξ= =, ξ U ξ= = 1. Case b = : exact stationary solution U := ξ + ξ2 2 ( ground state ). Case b : Look for asymptotic expansion in the form U = U + bt 1 + b 2 T 2 + Then T 1 is given by b ξξ T 1 = A(U ) + 1 T 1 = ξ4 48. What about T 2?

31 Finding the ODE on b Rule: choice of the approx. solution with the least possible growth. Remainder for U 1 := U + bt 1 : A(U 1 ) ξξ U ( 4 = α 5 b s + 13 ) 1 b2 ξ α ( b s + b 2) ξ 6 + α 2 b ( bs + b 2) ξ 8. 5 Choice of b such that the ξ 6 term disappears: b s + b 2 =. Ansatz: in the algorithm defining T N, replace every occurrence of b s by b 2. Consequence: setting U N := U + bt b N T N, U U 2 (b s + b 2 )(c 7 ξ 7 + c 8 ξ 8 ) near ξ =. U U N b s + b 2 for N 2.

32 Finding the ODE on b Rule: choice of the approx. solution with the least possible growth. Remainder for U 1 := U + bt 1 : A(U 1 ) ξξ U ( 4 = α 5 b s + 13 ) 1 b2 ξ α ( b s + b 2) ξ 6 + α 2 b ( bs + b 2) ξ 8. 5 Choice of b such that the ξ 6 term disappears: b s + b 2 =. Ansatz: in the algorithm defining T N, replace every occurrence of b s by b 2. Consequence: setting U N := U + bt b N T N, U U 2 (b s + b 2 )(c 7 ξ 7 + c 8 ξ 8 ) near ξ =. U U N b s + b 2 for N 2.

33 Finding the ODE on b Rule: choice of the approx. solution with the least possible growth. Remainder for U 1 := U + bt 1 : A(U 1 ) ξξ U ( 4 = α 5 b s + 13 ) 1 b2 ξ α ( b s + b 2) ξ 6 + α 2 b ( bs + b 2) ξ 8. 5 Choice of b such that the ξ 6 term disappears: b s + b 2 =. Ansatz: in the algorithm defining T N, replace every occurrence of b s by b 2. Consequence: setting U N := U + bt b N T N, U U 2 (b s + b 2 )(c 7 ξ 7 + c 8 ξ 8 ) near ξ =. U U N b s + b 2 for N 2.

34 Plan Introduction Behaviour near separation: heuristics and formal results Main result and ideas The approximate solution Energy estimates Conclusion and perspectives

35 Obtaining stability estimates General idea: control (b s + b 2 ) 2 via an appropriate energy E(s) := U U N 2. Goal: prove that E(s) = O(s 4 η ) for some η >. (2) Starting point: write eq. on U U N for N large (N = 3). of the form s (U U N ) + = remainder terms. Error estimate: prove that de ds + α E(s) ρ(s). s In order to achieve (2), one needs: ρ(s) = O(s 5 η ): good approximate solution; α > 4: algebraic manipulations on the equation (R).

36 A transport-diffusion equation for U Define, for W L (R + ), ξ L U W := UW U ξ W = so that, if W (ξ) = O(ξ 2 ) near ξ =, L 1 U W = (U ξ ( ξ W ) U 2, U ξ ) W U 2. ξ Remark: L 1 U division by U ξ + ξ2 /2. Then (R) can be written as s U bu + b 2 ξ ξu L 1 U ( ξξu 1) =. Define L U := L 1 U ξξ: diffusion operator. Then, with V = U U N s V bv + b 2 ξ ξv L U V = R N.

37 A transport-diffusion equation for U Define, for W L (R + ), ξ L U W := UW U ξ W = so that, if W (ξ) = O(ξ 2 ) near ξ =, L 1 U W = (U ξ ( ξ W ) U 2, U ξ ) W U 2. ξ Remark: L 1 U division by U ξ + ξ2 /2. Then (R) can be written as s U bu + b 2 ξ ξu L 1 U ( ξξu 1) =. Define L U := L 1 U ξξ: diffusion operator. Then, with V = U U N s V bv + b 2 ξ ξv L U V = R N.

38 A transport-diffusion equation for U Define, for W L (R + ), ξ L U W := UW U ξ W = so that, if W (ξ) = O(ξ 2 ) near ξ =, L 1 U W = (U ξ ( ξ W ) U 2, U ξ ) W U 2. ξ Remark: L 1 U division by U ξ + ξ2 /2. Then (R) can be written as s U bu + b 2 ξ ξu L 1 U ( ξξu 1) =. Define L U := L 1 U ξξ: diffusion operator. Then, with V = U U N s V bv + b 2 ξ ξv L U V = R N.

39 Energy and dissipation terms V = U U N, s V bv + b 2 ξ ξv L U V = R N. Facts: 1. Estimates are almost linear (up so some commutators...) 2. V = U U N (b s + b 2 )(c 1 ξ 7 + c 2 ξ 8 ) for ξ 1; 3. L U is a diffusion operator (L U 1 U ξξ). Ideas: Differentiate equation/use weights/apply operator L U to make the zero-order + transport term positive: k ξ ( s V bv + b 2 ξ ξv ) ( = s + k 2 b + b 2 2 ξ ξ ) k ξ V. Compromise between control of (b s + b 2 ) 2 by energy/small remainder term/positivity of transport and diffusion... Energy E(s) := (L 2 U V ) ξ 2 H 1 (w) for some weight w.

40 Energy and dissipation terms V = U U N, s V bv + b 2 ξ ξv L U V = R N. Facts: 1. Estimates are almost linear (up so some commutators...) 2. V = U U N (b s + b 2 )(c 1 ξ 7 + c 2 ξ 8 ) for ξ 1; 3. L U is a diffusion operator (L U 1 U ξξ). Ideas: Differentiate equation/use weights/apply operator L U to make the zero-order + transport term positive: k ξ ( s V bv + b 2 ξ ξv ) ( = s + k 2 b + b 2 2 ξ ξ ) k ξ V. Compromise between control of (b s + b 2 ) 2 by energy/small remainder term/positivity of transport and diffusion... Energy E(s) := (L 2 U V ) ξ 2 H 1 (w) for some weight w.

41 Energy and dissipation terms V = U U N, s V bv + b 2 ξ ξv L U V = R N. Facts: 1. Estimates are almost linear (up so some commutators...) 2. V = U U N (b s + b 2 )(c 1 ξ 7 + c 2 ξ 8 ) for ξ 1; 3. L U is a diffusion operator (L U 1 U ξξ). Ideas: Differentiate equation/use weights/apply operator L U to make the zero-order + transport term positive: k ξ ( s V bv + b 2 ξ ξv ) ( = s + k 2 b + b 2 2 ξ ξ ) k ξ V. Compromise between control of (b s + b 2 ) 2 by energy/small remainder term/positivity of transport and diffusion... Energy E(s) := (L 2 U V ) ξ 2 H 1 (w) for some weight w.

42 Tools for the proof Weighted L 2 estimates; Commutator estimates; L estimates coming from maximum principle (sub-super solutions); Bootstrap argument.

43 Conclusion and perspectives Plan Introduction Behaviour near separation: heuristics and formal results Main result and ideas Conclusion and perspectives

44 Conclusion and perspectives Summary Proof of separation for the stationary Prandtl equation in the case of adverse pressure gradient (g(x) = 1); Computation of a self-similar rate compatible with Landau s predictions: Y u Y = x x; Quantitative error estimates between true solution and approximate solution (in weighted H s spaces); Construction of an approximate solution, ODE on the separation rate: relies on arguments close to singularity formation for the nonlinear Schrödinger equation. Energy estimates rely heavily on the structure of the equation, and need to be combined with maximum principle techniques.

45 Conclusion and perspectives Perspectives Other (unstable) separation rates? Better description of the solution in the zone Y (x x) 1/4 ( ξ s 1/4 ); Higher dimensions? What happens after separation? 1. Both turbulent and laminar regimes are possible... But turbulent regimes are out of reach for the time being. 2. The validity of the Prandtl system after separation is far from clear... Thank you for your attention!

On the well-posedness of the Prandtl boundary layer equation

On the well-posedness of the Prandtl boundary layer equation Vlad Vicol Department of Mathematics, The University of Chicago Incompressible Fluids, Turbulence and Mixing In honor of Peter Constantin s